# American Institute of Mathematical Sciences

August  2015, 9(3): 375-390. doi: 10.3934/amc.2015.9.375

## Some new results on cross correlation of $p$-ary $m$-sequence and its decimated sequence

 1 School of Mathemetics & Computation Science, Anqing Normal University, Anqing, Anhui 246133, China 2 Department of Mathematic and Statistics, Centtal China Normal University, Wuhan, Hubei 430079, China 3 Department of Mathematic Sciences, Yangzhou University, Yangzhou, Jiangsu 225002, China, China

Received  September 2014 Published  July 2015

Let $p$ be an odd prime, $n=2m$, and $n/\gcd(k,n)$ be odd. In this paper, we study the cross correlation between a $p$-ary $m$-sequence $(s_{t})$ of period $p^{n}-1$ and its decimated sequence $(s_{dt})$ where $d$ satisfies $d(p^k+1)\equiv p^m+1 \pmod {p^n-1}$. Our results show that the cross-correlation function is six-valued and the distribution of the cross correlation is also completely determined.
Citation: Wenbing Chen, Jinquan Luo, Yuansheng Tang, Quanquan Liu. Some new results on cross correlation of $p$-ary $m$-sequence and its decimated sequence. Advances in Mathematics of Communications, 2015, 9 (3) : 375-390. doi: 10.3934/amc.2015.9.375
##### References:
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##### References:
 [1] A. W. Bluher, On $x^{q+1}+ax+b$,, Finite Fields Appl., 10 (2004), 285.  doi: 10.1016/j.ffa.2003.08.004.  Google Scholar [2] W. Chen, J. Luo and Y. Tang, Exponential sums from half quadratic forms and its applications,, in Proc. ISIT'14, (2014), 3145.   Google Scholar [3] S. T. Choi, J. S. No and H. Chung, On the cross-correlation of a ternary m-sequence of period $3^{4k+2}-1$ and its decimated sequence by $(3^{2k+1}+1)^{2}$ over 8,, in Proc. ISIT'10, (2010), 1268.   Google Scholar [4] S. T. Choi, J. S. No and H. Chung, On the cross-correlation of a $p$-ary m-sequence of period $p^{2m}-1$ and its decimated sequence by $\frac{(p^m+1)^2}{2(p+1)}$,, IEEE Trans. Inf. Theory, 58 (2012), 1873.  doi: 10.1109/TIT.2011.2177573.  Google Scholar [5] H. Dobbertin, P. Felke and T. Helleseth, Niho type cross correlation functions via Dickson polynomials and Kloosterman sums,, IEEE Trans. Inf. Theory, 52 (2006), 613.  doi: 10.1109/TIT.2005.862094.  Google Scholar [6] T. Helleseth, Some results about the cross-correlation function between two maximal-linear sequence,, Discrete Math., 16 (1976), 209.   Google Scholar [7] R. Lidl and H. Niederreiter, Finite Fields,, Addison-Wesley, (1983).   Google Scholar [8] J. Luo and K. Feng, On the weight distribution of two classes of cyclic codes,, IEEE Trans. Inf. Theory, 54 (2008), 5332.  doi: 10.1109/TIT.2008.2006424.  Google Scholar [9] J. Luo and K. Feng, Cyclic codes and sequences from generalized Coulter-Matthews function,, IEEE Trans. Inf. Theory, 54 (2008), 5345.  doi: 10.1109/TIT.2008.2006394.  Google Scholar [10] J. Luo, T. Helleseth and A. Kholosha, Two nonbinary sequences with six-valued cross correlation,, in Proc. IWSDA'11, (2011), 44.   Google Scholar [11] J, Luo, Y. Tang and H. Wang, Exponential sums, cyclic codes and sequences: the odd characteristic Kasami case,, preprint, ().   Google Scholar [12] G. J. Ness, T. Helleseth and A. Kholosha, On the correlation distribution of the Coulter-Matthews decimation,, IEEE Trans. Inf. Theory, 52 (2006), 2241.  doi: 10.1109/TIT.2006.872857.  Google Scholar [13] E. Y. Seo, Y. S. Kim, J. S. No and D. J. Shin, Cross-correlation distribution of p-ary m-sequence of period $p^{4k}-1$ and its decimated sequences by $(\frac{p^{2k}+1}{2})^{2}$,, IEEE Trans. Inf. Theory, 54 (2008), 3140.  doi: 10.1109/TIT.2008.924694.  Google Scholar [14] Y. Sun, Z. Wang, H. Li and T. Yan, The cross-correlation distribution of a $p$-ary m-sequence of period $p^{2k}-1$ and its decimated sequence by $\frac{(p^k+1)^2}{2(p^e+1)}$,, Adv. Math. Commun., 7 (2013), 409.  doi: 10.3934/amc.2013.7.409.  Google Scholar [15] Y. Xia, C. Li, X. Zeng and T. Helleseth, Some results on cross-correlation distribution between a $p$-ary $m$-sequence and its decimated sequences,, IEEE Trans. Inf. Theory, 60 (2014), 7368.  doi: 10.1109/TIT.2014.2350775.  Google Scholar
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